ARTICLE IN PRESS Robotics and Computer-Integrated Manufacturing 25 (2009) 630– 640

Contents lists available at ScienceDirect

Robotics and Computer-Integrated Manufacturing journal homepage: www.elsevier.com/locate/rcim

Dynamic modeling and robust control of a 3-PRC translational parallel kinematic machine Yangmin Li , Qingsong Xu ´s Pereira, Taipa, Macao SAR, China Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Av. Padre Toma

a r t i c l e in f o

a b s t r a c t

Article history: Received 12 September 2007 Received in revised form 7 May 2008 Accepted 20 May 2008

The dynamic modeling and robust control for a three-prismatic-revolute-cylindrical (3-PRC) parallel kinematic machine (PKM) with translational motion have been investigated in this paper. By introducing a mass distribution factor, the simplified dynamic equations have been derived via the virtual work principle and validated on a virtual prototype with the ADAMS software package. Based upon the established model, three dynamics controllers have been attempted on the 3-PRC PKM. The intuitive co-simulations with the combination of MATLAB/Simulink and ADAMS show that the control performance of neither inverse dynamics control nor robust inverse dynamics control is satisfactory in the presence of parametric uncertainties in PKM dynamics. On the contrary, the controller based on the passivity-based robust control scheme is more suitable for tracking control of the PKM in terms of both control performances and controller design procedures. The results presented in the paper provide a sound base for both the mechanical system design and control system design of a 3-PRC PKM. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Parallel robots Inverse dynamics Robust control Tracking control

1. Introduction In recent years, the research and development activities on parallel kinematic machines (PKMs) have been accelerated along with the evolution on both hardware and software of computer technology since the latter provides a strong support for researchers and engineers in robotics community. A PKM generally consists of a moving platform that is connected to a fixed base through more than one limbs together. Comparing with a traditional serial manipulator that is constituted with rigid-body links and joints connected in serial, a PKM possesses a more rigid structure and better payload carrying ability. Thus, it is more suitable for situations where high precision, stiffness, velocity, and heavy load-carrying are required within a restricted workspace [1]. Up to now, most of the investigations can be found on kinematics issues of PKM [2–5], while relatively few researches can be referred to the dynamics control of PKM. Ordinarily, there are two problems in PKM dynamics, namely the inverse and forward dynamics, where the former solves the actuation forces of actuators once the trajectories (in either joint space or task space) are planned, and the latter deals with the output motion of the PKM when the actuation forces are given. The inverse dynamics can be used for the design of a dynamics controller, whereas the

 Corresponding author. Tel.: +853 83974464; fax: +853 28838314.

E-mail address: [email protected] (Y. Li). 0736-5845/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2008.05.006

forward dynamics may be adopted for dynamics simulation of a PKM which can also be conducted by resorting to effective dynamics software packages such as ADAMS, DADS, and RecurDyn, etc. As far as the approaches to generate the PKM inverse dynamic model are concerned, most traditional methods are used, such as Newton–Euler formulation [6], Lagrangian formulation [7,8], virtual work principle [9,10], and some other methods [11]. With respect to the real-time control of a PKM, the goal of dynamic modeling is to establish an inverse dynamic model which is simple yet accurate enough to represent the PKM system. In general, once a PKM is designed and developed, its manipulation accuracy can be guaranteed by designing a proper controller. The inverse dynamics control (IDC) can produce a nice control performance provided that a full dynamic model of the PKM is used with all dynamic parameters known exactly [12]. Nevertheless, there will always exist uncertainties in the dynamic model due to the difficulty to accurately identify the model parameters, or the existence of unmodeled dynamics arising from the simplifications in modeling process. Hence, the ideal control performance of the IDC method will be degraded. To solve this problem, the robust control [13–16] and adaptive control [17–20] schemes can be adopted. Generally speaking, the robust controller is a fixed controller designed to meet the performance requirements over a given range of uncertainties, whereas an adaptive controller utilizes the manner of parameter estimation to adapt the computational model to the dynamic model on line. However, an adaptive controller that performs well in the presence of parametric uncertainties may not perform ideally in the case

ARTICLE IN PRESS Y. Li, Q. Xu / Robotics and Computer-Integrated Manufacturing 25 (2009) 630–640

of other types of uncertainties in terms of unmodeled dynamics or external disturbances. From this point of view, a robust controller can compensate more uncertainties than an adaptive one. Therefore, we employ a simplified dynamic model in conjunction with a robust controller to seek for an effective approach to deal with the control issues for a PKM in this research. In previous works of the authors, a 3-PRC (three-prismaticrevolute-cylindrical) PKM with relative simple structure was presented in [21] with its kinematic problems solved in details. After that, the stiffness modeling was performed, which showed that the overconstrained 3-PRC PKM could be converted into a non-overconstrained 3-CRC (three-cylindrical-revolute-cylindrical) PKM without any influences on its mobility and kinematics [22]. The objective of the current research is to establish the inverse dynamic model and implement the robust control for a 3-PRC PKM in case of dynamics uncertainties. With the addition of validating the dynamics model in ADAMS, we accomplish a cosimulation on a virtual prototype by combining MATLAB with ADAMS software to verify the designed controllers [23]. The virtual prototype provides a test bed via computer technology so as to verify control algorithm and enhance the design timely just before the development of a prototype. Besides, the co-simulation allows not only a visual view of the animation for the PKM behavior induced by the controller but also an inspection of whether the designed prototype and controllers are satisfactory or not for real developments. So, it is more intuitive than most of the existing simulations purely based on control algorithms. In the remainder of the paper, after a brief review of the 3-PRC PKM architecture and its kinematics in Section 2, the dynamic model is established and validated in Section 3. Afterwards, the IDC scheme is applied on the PKM in Section 4, and a robust IDC for the PKM is adopted in Section 5. However, it has been illustrated that the designed robust controller is not clearly superior to the former one, which motivates the design of a more effective one in Section 6 along with the performances validated through co-simulations. Finally, some concluding remarks are summarized in Section 7.

631

A2

Base platform y



A1

z

 x

C2 A3

P joint



R joint

C1

C3

D l20 10 l30 (B2') B2

v

w

C joint

B1 (B1')

P

B3 (B3')

u

Mobile platform

Fig. 2. Schematic representation of a 3-PRC PKM.

y z

Ai

ai

O



Mi

Ci

x Ni

didi0

p

lli0

Li

v

w

Bi P

bi'

Bi'

sisi0

u 2. Architecture and kinematics description The virtual prototype and schematic diagram for a 3-PRC PKM are shown in Figs. 1 and 2, respectively. The PKM consists of a moving platform, a fixed base, and three limbs with identical kinematic structure. Each limb connects the moving platform to

Fig. 1. A virtual prototype for a 3-PRC PKM.

Fig. 3. Representation of vectors.

the base by a P (prismatic) joint, a R (revolute) joint, and a C (cylindrical) joint in sequence, where the P joint is driven by a linear actuator assembled on the fixed base. Thus, the moving platform is attached to the base by three identical PRC linkages. Since the kinematics and mobility problems of the 3-PRC PKM have already been resolved in [21,22], we only review some useful results below which provide a base for the present research. To facilitate the analysis, as shown in Figs. 2 and 3, we assign a fixed Cartesian frame Ofx; y; zg at the centered point O of the fixed base, and a moving frame Pfu; v; wg on the triangle moving platform at the centered point P, along with the y- and v-axes perpendicular to the platform, and the x- and z-axes parallel to the u- and w-axes, respectively. Both DA1 A2 A3 and DB1 B2 B3 are assigned to be equilateral triangles so as to obtain a symmetric workspace of the manipulator. In addition, the ith limb C i Bi ði ¼ 1; 2; 3Þ with the length of l is connected to the moving platform at Bi which is a point on the axis of the ith C joint. B0i denotes the point on the moving platform that is coincident with the initial position of Bi , and the three points B0i lie on a circle of radius b. The three rails M i N i intersect one another at point D and intersect the x–z plane at points A1 , A2 , and A3 that lie on a circle of radius a. The sliders of P joints C i are restricted to move along the rails between Mi and Ni . Moreover, the axes of the R and C joints within the ith limb are parallel to each other to ensure only translational motion of the platform. Angle a is measured from the fixed base to

ARTICLE IN PRESS 632

Y. Li, Q. Xu / Robotics and Computer-Integrated Manufacturing 25 (2009) 630–640

rails M i N i and is defined as the layout angle of actuators. Without ! loss of! generality, let the z-axis point along A1 O, and the w-axis 0 along B1 P . Additionally, let dmax and smax denote the maximum stroke of linear actuators and passive C joints, respectively. The purpose of the inverse kinematics is to solve the actuated variables from a given position of the moving platform. Given a set of actuation inputs, the position of the moving platform can be solved by the forward kinematic analysis. With reference to Fig. 3, a vector-loop equation can be written for the ith limb as l li0 ¼ Li  di di0 ,

(1)

with the notation of

of the moving platform and the slider become m ¯ p ¼ mp þ 32wml ,

(6)

m ¯ s ¼ ms þ 12ð1  wÞml .

(7)

The simplification method for other types of PKM was reported for the dynamic modeling of the DELTA parallel robot [24], where it was directly assigned that one third of the passive leg mass was attached to the moving platform, i.e., w ¼ 13. In what follows, we will show that it is a different case for a 3-PRC PKM and a proper value of w will be determined based on dynamics simulation.

0

Li ¼ p þ bi þ si si0  ai ,

(2) ! where li0 is the unit vector along C i Bi , di represents the linear displacement of the ith actuated joint, di0 is the unit vector directing along rail M i Ni , si is the stroke of the ith C joint, and si0 denotes the unit vector parallel to the axes of the C and R joints of limb i. Substituting Eq. (2) into Eq. (1) and dot-multiplying both sides of the expression by si0 allows the derivation of si , i.e., si ¼ sTi0 p.

(3)

Additionally, differentiating both sides of Eq. (1) with respect to time along with a necessary calculation allows the derivation of the velocity equation: _ q_ ¼ J x,

(4) T _ _ _ where q_ ¼ ½d1 d2 d3  is a vector of actuated joint rates, x_ ¼ ½p_ x p_ y p_ z T ¼ ½vx vy vz T denotes the vector of linear velocities for the moving platform, and J ¼ J1 q Jx is defined as the Jacobian matrix of a 3-PRC PKM relating output velocities to the actuated joint rates, where 2 T 3 2 T 3 l10 l10 d10 0 0 6 T 7 6 l20 7 6 7 T 6 7 l20 d20 0 7 Jq ¼ 6 (5) 4 0 5 ; Jx ¼ 6 T 7 . 4 l30 5 T 0 0 l30 d30 33

33

3. Dynamic modeling and validation The main object of dynamic analysis for a PKM is to develop an inverse dynamic model, which enables the computation of the required actuator forces/torques when a desired trajectory is given. At the same time, for a real-time implementation of a dynamics controller, the model should be simple and accurate enough to represent the machine dynamics. In what follows, by introducing a mass distribution factor, we perform the dynamic modeling for a 3-PRC PKM using the virtual work principle and validate it through the dynamics simulation in ADAMS environment. 3.1. Simplification hypotheses Concerning a 3-PRC PKM, the complexity of its dynamics partly comes from the three moving legs. Since the legs can be manufactured with light materials, we can simplify the dynamics problem by defining a mass distribution factor w ð0owo1Þ for the legs. That is, the mass of each leg is divided into two portions and placed at its two extremities, i.e., one part with the proportion of w at its lower extremity (moving platform), and the other part ð1  wÞ at its upper extremity (the slider), and hence the rotational inertias of legs are neglected. Let mp , ms , and ml denote the masses for the moving platform, one slider, and one leg, respectively. Then, the equivalent masses

3.2. Dynamic modeling Assume that f ¼ ½f 1 f 2 f 3 T is a vector of actuator forces, dq ¼ ½dd1 dd2 dd3 T and dx ¼ ½dpx dpy dpz T are the vectors of virtual linear displacements for the sliders and the moving platform, respectively. Besides, let ds ¼ ½ds1 ds2 ds3 T be a vector for the virtual displacements of C joints with respect to the moving platform. Applying the virtual work principle allows the derivation of the following equation by neglecting the friction forces in passive R and C joints and assuming that there are no external forces exerted: T

f dq þ GTs dq  FTs dq þ GTp dx  ðFTp dx  FTl dsÞ ¼ 0,

(8)

where Gs ¼ ½m ¯ s gsa m ¯ s gsaT is the vector of gravity forces of ¯ s gsa m sliders with g denoting the gravity acceleration, Gp ¼ ½0  m ¯ p g 0T is the gravity force vector of the moving platform, Fs ¼ ½m ¯ s d€ 1 m ¯ s d€ 2 m ¯ s d€ 3 T describes a vector for the inertial forces of sliders, Fp ¼ ½m ¯ p p€ y m ¯ p p€ z T represents the vector of inertial ¯ p p€ x m forces of the moving platform, and Fl ¼ ½ðml Þ=2s€ 1 ðml Þ=2s€ 2 ðml Þ=2s€ 3 T is the vector of inertial forces for the legs of lower parts, respectively. In view of Eq. (3), we can derive that

ds ¼ s0 dx,

(9)

where 2

3 sT10 6 7 s0 ¼ 4 sT20 5 T s30

.

33

In addition, with reference to Eq. (4), we can obtain that _ x_ ¼ J1 q,

(10)

which leads to

dx ¼ J1 dq.

(11)

Substituting Eqs. (9) and (11) into Eq. (8), yields T

ðf þ GTs  FTs þ GTp J1  FTp J1 þ Fl s0 J1 Þdq ¼ 0.

(12)

Since Eq. (12) holds for any virtual displacements dq, we have f ¼ Fs þ JT Fp  JT sT0 Fl  Gs  JT Gp .

(13)

Next, substituting the inertial forces into Eq. (13), results in the computed forces: f ¼ Ms q€ þ JT Mp x€  JT sT0 Ml s0 x€  Gs  JT Gp ,

(14)

ARTICLE IN PRESS Y. Li, Q. Xu / Robotics and Computer-Integrated Manufacturing 25 (2009) 630–640

where 2

3

2

m ¯p 0 m ¯s 0 6 7 6 7 6 0 0 m 0 ¯ ; M ¼ Ms ¼ 6 s p 4 5 4 0 0 0 m ¯s 2 3 ml =2 0 0 6 7 0 ml =2 0 7. Ml ¼ 6 4 5 0 0 ml =2

0 m ¯p 0

633

Table 1 Kinematic and dynamic parameters of a 3-PRC PKM

3

0 7 0 7, 5 m ¯p

Parameter

Value

Parameter

Value

a b l mp ms

0.336 m 0.152 m 0.400 m 3.982 kg 1.297 kg

a dmax smax ml g

30:0 0.310 m 0.250 m 0.906 kg

−15

In addition, differentiating Eqs. (4) and (10) with respect to time, respectively, leads to _ q€ ¼ J x€ þ _J x,

(15)

1 _ x€ ¼ J1 q€ þ _J q.

(16)

9:807 m=s2

And then, substituting Eq. (16) into Eq. (14), allows the generation of dynamic model for a 3-PRC PKM expressed in joint space: _ q_ þ GðqÞ, f ¼ MðqÞq€ þ Cðq; qÞ

(17)

where MðqÞ ¼ Ms þ JT Mp J1  JT sT0 Ml s0 J1 , _ ¼ ðJT Mp  JT sT0 Ml s0 Þ_J Cðq; qÞ

1

,

(18a)

Actuation force (N)

−20

−25

−30 f2

(18b)

Dynamic model f1

f3

ADAMS output

−35 GðqÞ ¼ Gs  JT Gp .

(18c)

On the other hand, substituting Eq. (15) into Eq. (14), results in the dynamic model of a 3-PRC PKM described in task space: T

_ x_ þ Gx ðxÞ, f x ¼ J f ¼ Mx ðxÞx€ þ Cx ðx; xÞ

(19)

with T

Mx ðxÞ ¼ J Ms J þ Mp 

sT0 Ml s0 ,

(20a)

_ ¼ JT Ms _J, Cx ðx; xÞ

(20b)

Gx ðxÞ ¼ JT Gs  Gp ,

(20c)

3

33

is a where x 2 R denotes the controlled variables, Mx ðxÞ 2 R _ 2 R33 represents the matrix of centrifugal inertial matrix, Cx ðx; xÞ and Coriolis forces, and Gx ðxÞ 2 R3 is the vector of gravity forces. In view of Eqs. (18) and (20), it is observed that the dynamic model in task space is less complicated than that in joint space, hence the model of Eq. (19) is adopted for the following dynamics analysis and control purposes. It can be shown that the dynamic model possesses two noticeable features, i.e., the inertial matrix _ x  2Cx satisfies the skew Mx is symmetric positive definite and M symmetry property. The main problem in practical implementation of task space control for a PKM comes from the acquisition of the output posture of the PKM moving platform. For real implementation, employing redundant sensors in measurement will be easy to realize nowadays. In the current simulation studies, the forward kinematics for a 3-PRC PKM is solved on-line by resorting to the Newton–Raphson method thanks to the high performance of the computer hardware. 3.3. Validation on dynamic model In ADAMS, a virtual prototype for a 3-PRC PKM with kinematic and dynamic parameters described in Table 1 has been created. The established dynamic model for the PKM has been verified by employing the ADAMS software package via a simulation study carried out below.

0

0.5

1 Time (s)

1.5

2

Fig. 4. Actuator forces obtained by dynamic formulation and ADAMS simulation in case of w ¼ 0:5.

Let the moving platform track a helical trajectory along the y-axis in the reference frame, i.e.,   p  3 px ¼ 0:05 sin t cos pt , (21a) 2 2 h p i py ¼ h0  0:025 1  cos t , 2 pz ¼ 0:05 cos

  p  3 t sin pt , 2 2

(21b)

(21c)

where t is the time variable in unit of second, and px , py , and pz are in units of meters, with h0 ¼ 0:3552 m denoting the home position height of the moving platform. For the simulation, the input displacements of linear actuators are solved via the inverse kinematics, and then exported to ADAMS to drive the PKM following the trajectory in Eq. (21). The actuation forces of the three actuators for w ¼ 0:5 are illustrated in Fig. 4, which can be observed that there are deviations between the two approaches due to the introduced hypotheses. Since the mass distribution factor w varies from 0 to 1, it is necessary to determine its value to get an optimal dynamics simplification. With the variation of the factor w, the error range distributions for the actuator forces obtained by the dynamic model with comparison to the ADAMS output are illustrated in Fig. 5. It is observed that the distribution of the actuation error range of the 3PRC PKM is the best in case of w ¼ 0:58 since the actuation force error reaches to the minimum value and the errors are almost symmetrically distributed about the zero mean value. Therefore, the optimal factor w ¼ 0:58 rather than w ¼ 13 is designed for the 3-PRC PKM in this research. Under such a case, the deviations of the computed actuation forces with respect to the ADAMS simulation results are plotted in Fig. 6, which reveals that the computed force errors of the dynamic model are within only 1:25%.

ARTICLE IN PRESS 634

Y. Li, Q. Xu / Robotics and Computer-Integrated Manufacturing 25 (2009) 630–640

Actuator force error range (%)

equations into equivalent linear and decoupled second-order systems [12].

5 4.1. IDC algorithm The dynamics control in task space utilizing the IDC method is implemented for a 3-PRC PKM in what follows. Firstly, the task space dynamic model in Eq. (19) can be rewritten into the form:

0

_ f x ¼ Mx ðxÞx€ þ Hx ðx; xÞ,

(22)

_ ¼ Cx ðx; xÞ _ x_ þ GðxÞ. _ where Hx ðx; xÞ Fig. 7 depicts a block diagram of the IDC scheme with proportional-derivative (PD) feedback. Assuming that there are no external disturbances, then the manipulator is actuated by the actuation forces described in the task space:

−5

_ f x ¼ Mx ðxÞu þ Hx ðx; xÞ,

−10 0

0.2

0.4 0.6 Mass factor w

0.8

1

Fig. 5. Distribution of the actuator force error range versus the mass factor w.

(23)

where u is an input signal vector in the form of acceleration. Combining Eq. (22) with Eq. (23) results in the following linear second-order system x€ ¼ u,

Actuation force deviation (%)

which indicates that the system of Eq. (23) under control in Eq. (24) is linear and decoupled with respect to the input vector u. And the reference signal can be defined according to the following algorithm:

Δf1

1

(24)

Δf2 Δf3

0.5

r ¼ x€ d þ KD x_ d þ KP xd ,

(25)

where xd denotes the desired trajectory for the moving platform, and the symmetric positive definite feedback gain matrices are chosen as the diagonal form:

0

KP ¼ diagfK P g ¼ diagfo2n g,

(26)

KD ¼ diagfK D g ¼ diagf2zon g,

(27)

−0.5

−1 0

0.5

1 Time (s)

1.5

2

Fig. 6. Deviations of computed actuation forces with respect to simulation results from ADAMS in case of w ¼ 0:58.

From the simulation results, we can see that the introduced simplification hypotheses are reasonable for the dynamic model of a 3-PRC PKM. Due to the simple architecture of the established dynamic model, it can be adopted for dynamics control of the PKM. It should be noted that different trajectories lead to different optimal values for the mass distribution factor w due to the variations on velocities and accelerations. Moreover, the mass factor is also leg dependant, which is different from leg to leg even for the same trajectory. This phenomenon can be observed by an insightful view of Fig. 6, which indicates that the optimal mass distribution factor w ¼ 0:58 for the PKM is actually optimized for leg 3 in terms of error distributions around corresponding mean values. For a more detailed study, different mass factors should be considered for different legs to obtain a more accuracy result, which is remained for our future research.

which indicates that the ith (i ¼ 1; 2, and 3) component of reference r influences only the ith DOF motion of the PKM with a natural frequency on and a damping ratio z. The acceleration input signal in Eq. (24) then becomes _ þ KP ðxd  xÞ. u ¼ x€ d þ KD ðx_ d  xÞ

Substituting Eq. (28) into Eq. (24), leads to a homogeneous second-order differential equation of errors: e€ þ KD e_ þ KP e ¼ 0,

As an important basis for dynamics control, the IDC scheme is based upon the transformation of the nonlinear dynamic

(29)

where e ¼ xd  x

(30)

is the vector of displacement tracking errors. It has been shown that the error system in Eq. (29) is asymptotically stable along with the positive definite matrices KP and KD .

D xd

xd 4. Inverse dynamics control

(28)

xd

u

fx

Mx(x)

KP

KD

e

e

f J-T

q 3-PRC PKM

q

Hx(x, x) x x

J-1 FK

Fig. 7. Block diagram of inverse dynamics control scheme in task space.

ARTICLE IN PRESS Y. Li, Q. Xu / Robotics and Computer-Integrated Manufacturing 25 (2009) 630–640

635

is 6 Hz. Then, the natural frequency of the control system can be designed as: on ¼ 2p  3 Hz ¼ 18:85 rad=s, which allows the determination of the feedback gains:

4.2. Simulation results and discussions A co-simulation on the virtual prototype of the 3-PRC PKM has been accomplished by combining MATLAB/Simulink with ADAMS. During the simulation, the control algorithm is executed under MATLAB to generate the command forces, which are then exported to ADAMS environment and applied to the actuators of the virtual prototype at each cycle of time. The resulted outputs of the virtual prototype (position and velocity of the moving platform) are measured by ‘‘sensors’’ in ADAMS and then fed back to the controller in MATLAB for calculation of the next command signal. In such a communication manner, the co-simulation proceeds until the end of time, and the input and output variables for the virtual prototype are illustrated in Fig. 8. The manipulator is commanded to track a trajectory defined in Eq. (21). In order to make the system response critically damped, the damping ratio is chosen as z ¼ 1:0 to eliminate overshooting of tracking errors. Moreover, to avoid exciting the structural oscillation and resonance of the PKM system, the undamped natural frequency on may be designed no more than one-half of the structural resonant frequency os of the PKM, i.e., on pos =2. Assume that the mechanical resonant frequency of the 3-PRC PKM

K P ¼ 355:3

and

K D ¼ 37:7.

(31)

The simulation results show that the above IDC approach results in almost zero steady-state errors. However, in real situation, the payload and dynamic parameters may not be exactly known. With the normal dynamic parameters of the 3-PRC PKM deviated 20% from the exact values, the control results are illustrated in Fig. 9. With the forward kinematics solved on-line by the Newton– Raphson iterative method, the simulation time is about 10 s running on a personal computer with Pentium-4 3.00-GHz CPU. The oscillation ranges of position and velocity control errors are, respectively, defined as the tolerances of steady-state position _ after the settling time t s in errors etol ðeÞ and velocity errors etol ðeÞ this paper. We use both the maximum control errors emax ðeÞ and _ and the tolerances of steady-state errors to describe the emax ðeÞ control performances of a designed controller. From Fig. 9, we can observe that the maximum control errors are emax ðeÞ ¼ 4:8 mm, _ ¼ 5:6 mm=s, and the tolerances of steady-state errors are emax ðeÞ _ ¼ 10:8 mm=s, respectively, after t s ¼ 0:3 s. etol ðeÞ ¼ 5:8 mm, etol ðeÞ

1 ADAMS_uout

1

d1 2

f1

U To Workspace

d2 3

2

Demux

MSCSoftware

Mux

d3 4

f2

ADAMS Plant

v1 5

3

v2

ADAMS_yout

6

f3 Mux

v3

Demux

Y To Workspace ADAMS_tout Clock

T To Workspace

Fig. 8. Topology of control block diagram for the virtual prototype.

Velocity error (m/s)

5 0 −5

Pz 0

0.5

Px

Py

1 Time (s) Actuation force (N)

Position error (m)

x 10−3

1.5

0.04 Vy

0.02 0 −0.02

Vz

Vx

−0.04 0

2

0.5

1 Time (s)

−15 −20 −25 −30 f1

−35 0

f3 0.5

f2 1 Time (s)

1.5

2

Fig. 9. Simulation results of IDC controller in the presence of 20% uncertainties on dynamic parameters.

1.5

2

ARTICLE IN PRESS 636

Y. Li, Q. Xu / Robotics and Computer-Integrated Manufacturing 25 (2009) 630–640

Although the IDC control performance can be enhanced by increasing the feedback gains, the gains are restricted by the natural frequency of the PKM. Thus the control performance is limited, which provides a motivation for the design of a robust controller in the following section.

By setting   e E¼ e_

(39)

as the state of the system, Eq. (38) can be written as E_ ¼ AE þ Bðg  rÞ, where "

5. Robust IDC scheme

0

I

KP

KD

(40)

#

  0 . I

5.1. Robust inverse dynamics control (RIDC) design

A¼

Referring to the dynamic equation (22), in the presence of uncertainties including modeling errors, unknown loads, and parameters measurement, the manipulator is actuated with the following joint forces expressed in the task space:

Since the matrices KP and KD are chosen so that A in Eq. (40) is a Hurwitz matrix, i.e., A has all its eigenvalues in the open left half of the complex plane, any symmetric positive definite matrix P can be chosen to give a unique solution Q satisfying the relationship:

b x ðxÞu þ H b x ðx; xÞ, _ fx ¼ M

(32)

b x ðx; xÞ b x ðxÞ and H _ where u is a new input vector to be determined, M denote the estimators of the inertial matrix Mx ðxÞ and the _ implemented in the controller, nonlinear coupling matrix Hx ðx; xÞ respectively. The errors of the estimates, i.e., the uncertainties, can be expressed by b x  Mx ; ex ¼ M M

ex ¼ H b x  Hx . H

(33)

Taking Eq. (32) as a nonlinear control law gives b x ðxÞu þ H b x ðx; xÞ, _ ¼M _ Mx ðxÞx€ þ Hx ðx; xÞ

(34)

x€ ¼ u þ

b ðM1 x Mx

 IÞu þ

b M1 x Hx

¼ u  g,

(35)

B¼

AT Q þ QA ¼ P,

(41)

(42)

where Q is symmetric positive definite as well. Defining the control law for r in accordance with [15]: 8 _ rðe; eÞ > > l; if klk4; > < klk r¼ > _ rðe; eÞ > > : l; if klkp;

(43)



where l ¼ BT QE and 40, it follows that the control input in Eq. (38) is continuous and the Lyapunov function candidate VðEÞ ¼ ET QE40;

which allows the generation of

;

8 Ea0,

(44)

_ satisfies Vo0 along the trajectories of the error system of Eq. (40). The proof procedure is similar to that in [12] and hence is omitted here.

where I denotes a 3  3 identity matrix and 1 b b g ¼ ðI  M1 x Mx Þu  Mx Hx .

(36)

In view of the position error expressed by Eq. (30), the velocity _ and the acceleration error can error can be written as e_ ¼ x_ d  x, be calculated as follows with the consideration of Eq. (35). e€ ¼ x€ d  u þ g

(37)

In order to compensate for the uncertainties, the following input vector is chosen: u ¼ x€ d þ KD e_ þ KP e þ r,

(38)

where the gain matrices KP and KD are, respectively, defined in Eqs. (26) and (27), and r is an additional item to be designed to guarantee robustness to the effects of uncertainties described by g in Eq. (36), and the control block diagram is shown in Fig. 10.

 (e,e) xd

u

KP

xd xd

D

e

KD

fx

Mˆ x (x, x)

f J-T

q 3-PRC PKM

Hˆx (x, x)

e x x

J-1 FK

Fig. 10. Block diagram of robust inverse dynamics control scheme.

q

5.2. Simulations and discussions In order to implement the RIDC scheme presented above, we have to determine several control parameters in terms of KP , KD , P, , and r. Generally, the greater the uncertainty is, the larger the positive scaler r is, and the constant  can be chosen as large as necessary to reduce or eliminate chattering. Additionally, it is _ and the observed that r is a function of tracking errors (e, e) calculation of r requires the determination of bounds associated with the desired tracking trajectory and the uncertainties including X M , MM , k, and g. A co-simulation is carried out to command the 3-PRC PKM tracking the same trajectory described by Eq. (21). With the normal dynamic parameters offset 20% from the real values of the PKM, the trajectory and uncertainty bounds can be computed as: X M ¼ 1:24, M M ¼ 0:23, k ¼ 30:06, and g ¼ 0:85. In addition, the feedback gains are designed according to the natural frequency of the system as described in Eq. (31), and other control parameters are chosen as: P ¼ diagf10g and  ¼ 0:2. The simulation results are illustrated in Fig. 11, which shows that after t s ¼ 0:8 s, the control performances are emax ðeÞ ¼ 4:5 mm, _ ¼ 3:6 mm=s, etol ðeÞ ¼ 5:3 mm, and etol ðeÞ _ ¼ 6:8 mm=s, reemax ðeÞ spectively. Comparing the control results with those of the IDC, we can conclude that the reduction of control errors is not very obvious, while the settling time is increased by almost 2 times. So, the control performance of the designed robust controller is not clearly superior to the former inverse dynamics controller. By changing the values of design parameters, we can observe that the increasing values of some parameters may lead to less steady-state errors. For example, an increase of MM by 5 times, i.e., M M ¼ 1:38, leads to the control performances of emax ðeÞ ¼ _ ¼ 3:2 mm, etol ðeÞ ¼ 4:7 mm, and etol ðeÞ _ ¼ 4:6 mm=s, 3:9 mm, emax ðeÞ

ARTICLE IN PRESS

x 10−3

Velocity error (m/s)

Position error (m)

Y. Li, Q. Xu / Robotics and Computer-Integrated Manufacturing 25 (2009) 630–640

5 0 −5

Pz 0

Px

0.5

Py

1

1.5

637

0.04 Vy 0.02 0 −0.02

Vz

Vx

−0.04

2

0

0.5

1

1.5

2

1.5

2

Time (s)

Time (s) Actuation force (N)

−15 −20 −25 −30 f1

−35 0

f3

f2

0.5

1

1.5

2

Time (s) Fig. 11. Simulation results of RIDC controller.

Velocity error (m/s)

Position error (m)

x 10−3 5 0 −5

Pz 0

Py

Px

0.5

1

1.5

0.04 Vy 0.02 0 −0.02

Vz

Vx

−0.04 0

2

0.5

1 Time (s)

Actuation force (N)

Time (s) −15 −20 −25 −30 f1

−35 0

f3 0.5

f2 1 Time (s)

1.5

2

Fig. 12. Simulation results of RIDC controller with increased control parameters.

respectively, as depicted in Fig. 12. Thus, the maximum control errors on displacement and velocity tracking have been greatly reduced by 19% and 43%, and the error tolerances have been reduced by 19% and 57%, respectively, with respect to the IDC results. However, the control errors are reduced at the expense of a long settling time ðt s ¼ 1:6 sÞ to reach the steady state for all of the tracking errors, that is more than 4 times longer than the settling time in the IDC method. A further increase of control parameters will reduce the control errors even more at the sacrifice of longer settling time to arrive at the steady state, and induce more severe initial oscillations of the control errors at the same time. Hence, the design parameters cannot be increased infinitely, and the performance of the RIDC controller is limited. Besides, an overview of the design procedure for the robust controller reveals that there are as many as eight control parameters and uncertainty bounds to be determined in advance. This motivates us to seek a more simple controller for an effective control of a 3-PRC PKM in the presence of parameters uncertainties in the following section.

6. Passivity-based robust control scheme In this section, we attempt an alternative robust controller for a 3-PRC PKM based on the passivity or skew symmetry property of the dynamic equations. Making use of both the skew symmetry property and linearity in the parameters, the adopted control algorithm allows the design of a controller with reduced burden on the determination of uncertainty bounds.

6.1. Passivity-based robust control (PBRC) design In view of the linear parametrization property of the dynamic model of a manipulator, the dynamic equation (19) can be linearized as _ x_ þ Gx ðxÞ ¼ Yðx; x; _ xÞ € h ¼ fx, Mx ðxÞx€ þ Cx ðx; xÞ

(45)

where h is a m  1 vector of constant parameters and Y denotes a n  m matrix which is a regressor function of position, velocity and acceleration for the moving platform of the 3-PRC PKM. By

ARTICLE IN PRESS 638

Y. Li, Q. Xu / Robotics and Computer-Integrated Manufacturing 25 (2009) 630–640

the robust term n can be designed as follows [13]: 8 d > > > < kkkk if kkk4; n¼ > d > > if kkkp; : k

setting

h ¼ ½m ¯s

m ¯p

ml T ,

(46)

then a 3  3 regressor matrix Y can be found, whose components are all functions of x and its derivatives do not contain any dynamic parameters. For the sake of tracking the desired trajectory xd, the control input is designed as follows: b x ðx; xÞv b x ðxÞ þ Kr, b x ðxÞa þ C _ þG fx ¼ M



where k ¼ YT r and 40. Defining a Lyapunov function candidate for the system in Eq. (54) as

(47)

V ¼ 12rT Mx ðxÞr þ eT KKe,

where the quantities a; v, and r are defined as v ¼ x_ d þ Ke,

(48)

_ a ¼ v_ ¼ x€ d þ Ke,

(49)

r ¼ v  x_ ¼ e_ þ Ke,

(50)

6.2. Simulation results and discussions From the above design process of the control scheme, it can be seen that there are only four control parameters to be designed, which are the constants K, K, d, and . A comparison of the PBRC controller with the former designed RIDC one reveals that finding a constant bound d for the constant vector e y is much simpler than seeking for a time-varying bound r for g in Eq. (36). Because the bound d relies only on the inertia parameters of the PKM, while r depends on the state error, the desired trajectory, with bx the addition of some assumptions on the estimated matrices M b x. and H For the simulation study, with the normal dynamic parameters of the 3-PRC PKM deviated 20% from the exact values, it is easy to calculate: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi khk ¼ ð0:2m (58) ¯ s Þ2 þ ð0:2m ¯ p Þ2 þ ð0:2ml Þ2 p1:17,

(51)

h denotes the estimate for the parameter vector h and can where b be described by b h ¼ h0 þ n,

(52)

with h0 denoting a vector of fixed normal parameters and n being a robust term to be determined. Combining Eq. (47) with Eq. (45) allows the derivation of b  hÞ. _ r_ þ Kr ¼ Yðh Mx ðxÞr€ þ Cx ðx; xÞ

(53)

h ¼ h0  h be a constant vector describing the parametric Assign e uncertainty in the system. Then, Eq. (53) becomes e þ nÞ. _ r_ þ Kr ¼ Yðx; x; _ a; vÞðh Mx ðxÞr€ þ Cx ðx; xÞ

hence the uncertainty bound is designed as d ¼ 1:17. In addition, the gain matrices are selected as: K ¼ diagf150g and K ¼ diagf200g, respectively. The parameter  is chosen as 0.5, which results in almost no chattering and a quick response with the _ ¼ maximum control parameters of emax ðeÞ ¼ 1:4 mm and emax ðeÞ 3:7 mm=s, and steady-state error tolerances of etol ðeÞ ¼ 1:5 mm

(54)

In the case that the uncertainty is bounded by a constant dX0, i.e.,

hk ¼ kh0  hkpd, ke

(57)

it can be demonstrated that the closed-loop system is uniformly ultimately bounded [13] under control law of Eq. (51). The block diagram for the passivity-based robust control is illustrated in Fig. 13. The guideline for the design parameter of  lies in that a smaller  will lead to smaller tracking errors while the elimination or reduction of chattering requires a larger . Hence, a design tradeoff should be made to yield a better control performance.

with K and K representing diagonal constant matrices of positive gains. Taking into consideration the linearity property of the dynamic model, the control in Eq. (47) then becomes b þ Kr, _ a; vÞh f x ¼ Yðx; x;

(55)

D xd

xd

a

e

q

fx

Y (x,x,a,v) ˆ

f J-T

3-PRC PKM

Λ r K v

xd

e

(56)

Λ x x

J-1 FK

Fig. 13. Block diagram of passivity-based robust control scheme.

q

ARTICLE IN PRESS Y. Li, Q. Xu / Robotics and Computer-Integrated Manufacturing 25 (2009) 630–640

Velocity error (m/s)

Position error (m)

x 10−3 Py

5 0 −5

Pz 0

0.5

Px 1

1.5

0.04 Vy

0.02 0 −0.02

Vx

Vz

−0.04 0

2

0.5

1

Time (s) Actuation force (N)

639

1.5

2

0.6

0.8

Time (s)

−15 −20 −25 −30 f3

f1

−35 0

0.5

f2 1 Time (s)

1.5

2

Fig. 14. Simulation results of PBRC controller. Table 2 Control results from three types of controllers

1.6

t s (s)

emax ðeÞ (mm)

etol ðeÞ (mm)

_ emax ðeÞ (mm/s)

IDC RIDC (1) RIDC (2) PBRC

0.3 0.8 1.6 0.1

4.8 4.5 3.9 1.4

5.8 5.3 4.7 1.5

5.6 3.6 3.2 3.7

_ (mm/s) etol ðeÞ

10.8 6.8 4.6 7.4

_ ¼ 7:4 mm=s, respectively, after t s ¼ 0:1 s, as shown in and etol ðeÞ Fig. 14. For the convenience of comparison, the control results from three types of controllers are elaborated in Table 2. Comparing the PBRC control results with those of the IDC, we can see that the maximum control errors for the displacement and velocity tracking have been reduced by 71% and 34%, respectively. Comparing the current results with those from the RIDC with increased control parameters, it is observed that the settling time has been significantly shortened, and both the maximum control error and steady-state error tolerance for the displacement trajectory have been significantly reduced by 64% and 68%, respectively, although the velocity tracking errors have been increased by certain magnitudes. In most cases, given the desired trajectory of the moving platform of the PKM, the control errors for the displacement tracking are the most concerned factor. Overall, the designed PBRC is more practical than the RIDC controller in virtue of simple design procedures and good control performances. Besides, in view of Table 2, we can observe that the maximum displacement tracking errors from the dynamics control are a few millimeters, which are relatively larger than conventional kinematics-based motion control approach in a general sense. However, the control errors are obtained by a relatively large offset of 20% from the true parameters of the PKM dynamics actually. Under such a worse situation, the kinematics-based controller may probably lead to an even larger control error. Generally, the smaller the uncertainties of dynamic parameters are, the smaller the control errors are. Furthermore, since the implemented PBRC controller allows certain tolerance on the accuracy of the dynamic parameters, it is interesting to have a knowledge concerning the effects of mass distribution factor ðwÞ on the control performance in addition to the actuator forces. For instance, as the increasing of the mass factor from 0 to 1, the control performances for the aforementioned

Control performance (mm)

Controller

x 10−3

1.5 emax etol 1.4

1.3 0

0.2

0.4

1

Mass factor w Fig. 15. Control performance versus the mass factor w.

trajectory in terms of the maximum and tolerance of position errors are shown in Fig. 15. A comparison of the variation tendencies as shown in Figs. 5 and 15 reveals that the mass factor has similar tendency of effects on actuator forces and control performances. However, as reflected by the variation range of the maximum and tolerance of position errors (emax and etol ) within 0:1 mm, the influence of mass factor on control performances is trival than that on the actuator forces. This in part demonstrates the effectiveness of the implemented robust control scheme. Even so, from application point of view, it is necessary to optimize the mass factor with respect to control performances for various trajectories within the workspace of the PKM. Such a meaningful study is expected and planned in the next step of our future research.

7. Conclusions In this paper, the inverse dynamics modeling and robust controllers design for a 3-PRC PKM have been conducted. By optimizing the mass distribution factor concerning the legs, the simplified dynamic model of the PKM has been established and

ARTICLE IN PRESS 640

Y. Li, Q. Xu / Robotics and Computer-Integrated Manufacturing 25 (2009) 630–640

validated on a virtual prototype created in ADAMS environment. Based upon the derived model, three types of controllers in terms of inverse dynamics control (IDC), robust inverse dynamics control (RIDC), and passivity-based robust control (PBRC) in task space have been implemented on the virtual prototype through the co-simulation with MATLAB/Simulink and ADAMS. The simulation results show that the control performances of IDC are degraded if the normal parameters of the PKM are some degree of offsets from the exact values. It has been revealed that the performance of RIDC controller is not obviously superior to the IDC one, and the improvement of its control performance is at the cost of long settling time, which is not preferred for practical applications. In contrast, the control performances of PBRC controller have been significantly improved with comparison to the two former controllers, which are reflected by the fact that the maximum control errors of displacement tracking have been reduced by 71% and 64% with respect to the IDC and RIDC controllers, respectively, along with a shortened settling time. Moreover, the design process of a PBRC controller is greatly simplified with comparison to the RIDC controller. The analysis procedure also exhibits that a controller cannot be arbitrarily applied to a PKM without a careful selection. The main contribution of this paper is the establishment of simplified inverse dynamic model for a 3-PRC PKM and application of a suitable robust controller for trajectory tracking in the presence of dynamics uncertainties. The obtained results are helpful for the mechatronic design and development of a 3-PRC PKM. In the future work, the treatment of actuator’s dynamics and frictions in passive joints will be conducted once a physical prototype is developed. Moreover, the design and validation methodology presented in this paper can be extended to other types of PKM as well.

Acknowledgments The authors appreciate the fund support from the research committee of University of Macau under Grant no.: RG065/0607S/08T/LYM/FST and Macao Science and Technology Development Fund under Grant 069/2005/A. References [1] Merlet J-P. A generic trajectory verifier for the motion planning of parallel robots. ASME J Mech Des 2001;123(4):510–5.

[2] Liu X-J, Jin Z-L, Gao F. Optimum design of 3-DOF spherical parallel manipulators with respect to the conditioning and stiffness indices. Mech Mach Theory 2000;35(9):1257–67. [3] Yang G, Chen I-M, Lin W, Angeles J. Singularity analysis of three-legged parallel robots based on passive-joint velocities. IEEE Trans Robot Automat 2001;17(4):413–22. [4] Xi F, Zhang D, Mechefske CM, Lang SYT. Global kinetostatic modelling of tripod-based parallel kinematic machine. Mech Mach Theory 2004;39(4): 357–77. [5] Dai JS, Huang Z, Lipkin H. Mobility of overconstrained parallel mechanisms. ASME J Mech Des 2006;128(1):220–9. [6] Li Y-W, Wang J-S, Wang L-P, Liu X-J. Inverse dynamics and simulation of a 3-DOF spatial parallel manipulator. In: Proceedings of IEEE international conference on robotics and automation, 2003. p. 4092–7. [7] Di Gregorio R, Parenti-Castelli V. Dynamics of a class of parallel wrists. ASME J Mech Des 2004;126(3):436–41. [8] Li Y, Xu Q. Kinematics and inverse dynamics analysis for a general 3-PRS spatial parallel mechanism. Robotica 2005;23(2):219–29. [9] Wang J, Gosselin CM. A new approach for the dynamic analysis of parallel manipulators. Multibody Syst Dyn 1998;2(3):317–34. [10] Tsai L-W. Solving the inverse dynamics of a Stwart–Gough manipulator by the principle of virtual work. ASME J Mech Des 2000;122(1):3–9. [11] Khan WA, Krovi VN, Saha SK, Angeles J. Recursive kinematics and inverse dynamics for a planar 3R parallel manipulator. ASME J Dyn Syst Meas Control 2005;127(4):529–36. [12] Sciavicco L, Siciliano B. Modeling and control of robot manipulators. New York: The McGraw-Hill Companies, Inc.; 1996. [13] Spong MW. On the robust control of robot manipulators. IEEE Trans Automat Control 1992;37(11):1782–6. [14] Liu G, Goldenberg AA. Comparative study of robust saturation-based control of robot manipulators: analysis and experiments. Int J Robot Res 1996;15(5): 473–91. [15] Kim HS, Cho YM, Lee KI. Robust nonlinear task space control for 6 DOF parallel manipulator. Automatica 2005;41(9):1591–600. [16] Torresa S, Mendeza JA, Acostaa L, Becerra VM. On improving the performance in robust controllers for robot manipulators with parametric disturbances. Control Eng Practice 2007;15(5):557–66. [17] Feng G, Palaniswami M. Adaptive control of robot manipulators in task space. IEEE Trans Automat Control 1993;38(1):100–4. [18] Honegger M, Brega R, Schweitzer G. Application of a nonlinear adaptive controller to a 6 dof parallel manipulator. In: Proceedings of IEEE international conference on robotics and automation, 2000. p. 1930–5. [19] Liu Y, Li Y. Dynamic modeling and adaptive neural-fuzzy control for nonholonomic mobile manipulators moving on a slope. Int J Control Automat Syst 2006;4(2):197–203. [20] Yao B, Xu L. Output feedback adaptive robust control of uncertain linear systems with disturbances. ASME J Dyn Syst Meas Control 2006;128(4): 938–45. [21] Li Y, Xu Q. Kinematic analysis and design of a new 3-DOF translational parallel manipulator. ASME J Mech Des 2006;128(4):729–37. [22] Xu Q, Li Y. An investigation on mobility and stiffness of a 3-DOF translational parallel manipulator via screw theory. Robot Comput- Integr Manuf 2008; 24(3):402–14. [23] Callegari M, Palpacelli M-C, Principi M. Dynamics modelling and control of the 3-RCC translational platform. Mechatronics 2006;16(10):589–605. [24] Codourey A. Dynamic modeling and mass matrix evaluation of the DELTA parallel robot for axes decoupling control. In: Proceedings of IEEE/ RSJ international conference on intelligent robots and systems, 1996. p. 1211–8.